KILLED proof of input_3YXcv52Bt6.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 243 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 132 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 634 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] (64) CdtProblem (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) power(x, Nil) -> Cons(Nil, Nil) mult(x, Nil) -> Nil add0(x, Nil) -> x goal(x, y) -> power(x, y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) power(x, Nil) -> Cons(Nil, Nil) mult(x, Nil) -> Nil add0(x, Nil) -> x goal(x, y) -> power(x, y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) power(x, Nil) -> Cons(Nil, Nil) mult(x, Nil) -> Nil add0(x, Nil) -> x goal(x, y) -> power(x, y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1] mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1] add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1] power(x, Nil) -> Cons(Nil, Nil) [1] mult(x, Nil) -> Nil [1] add0(x, Nil) -> x [1] goal(x, y) -> power(x, y) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1] mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1] add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1] power(x, Nil) -> Cons(Nil, Nil) [1] mult(x, Nil) -> Nil [1] add0(x, Nil) -> x [1] goal(x, y) -> power(x, y) [1] The TRS has the following type information: power :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil mult :: Cons:Nil -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: goal_2 (c) The following functions are completely defined: power_2 mult_2 add0_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1] mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1] add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1] power(x, Nil) -> Cons(Nil, Nil) [1] mult(x, Nil) -> Nil [1] add0(x, Nil) -> x [1] goal(x, y) -> power(x, y) [1] The TRS has the following type information: power :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil mult :: Cons:Nil -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: power(x', Cons(x, Cons(x1, xs'))) -> mult(x', mult(x', power(x', xs'))) [2] power(x', Cons(x, Nil)) -> mult(x', Cons(Nil, Nil)) [2] mult(x', Cons(x, Cons(x2, xs''))) -> add0(x', add0(x', mult(x', xs''))) [2] mult(x', Cons(x, Nil)) -> add0(x', Nil) [2] add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1] power(x, Nil) -> Cons(Nil, Nil) [1] mult(x, Nil) -> Nil [1] add0(x, Nil) -> x [1] goal(x, y) -> power(x, y) [1] The TRS has the following type information: power :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil mult :: Cons:Nil -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(x', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' goal(z, z') -{ 1 }-> power(x, y) :|: x >= 0, y >= 0, z = x, z' = y mult(z, z') -{ 2 }-> add0(x', add0(x', mult(x', xs''))) :|: x' >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0, z = x' mult(z, z') -{ 2 }-> add0(x', 0) :|: x' >= 0, x >= 0, z' = 1 + x + 0, z = x' mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 power(z, z') -{ 2 }-> mult(x', mult(x', power(x', xs'))) :|: x1 >= 0, x' >= 0, x >= 0, xs' >= 0, z = x', z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(x', 1 + 0 + 0) :|: x' >= 0, x >= 0, z' = 1 + x + 0, z = x' power(z, z') -{ 1 }-> 1 + 0 + 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { add0 } { mult } { power } { goal } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {add0}, {mult}, {power}, {goal} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {add0}, {mult}, {power}, {goal} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: add0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + 2*z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {add0}, {mult}, {power}, {goal} Previous analysis results are: add0: runtime: ?, size: O(n^1) [z + 2*z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: add0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 + xs }-> 1 + (1 + 0 + 0) + s' :|: s' >= 0, s' <= z + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 3 }-> s :|: s >= 0, s <= z + 2 * 0, z >= 0, z' - 1 >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: mult after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 + xs }-> 1 + (1 + 0 + 0) + s' :|: s' >= 0, s' <= z + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 3 }-> s :|: s >= 0, s <= z + 2 * 0, z >= 0, z' - 1 >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z'] mult: runtime: ?, size: EXP ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: mult after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 + xs }-> 1 + (1 + 0 + 0) + s' :|: s' >= 0, s' <= z + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0 goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0 mult(z, z') -{ 3 }-> s :|: s >= 0, s <= z + 2 * 0, z >= 0, z' - 1 >= 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs') power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0 power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z'] mult: runtime: INF, size: EXP ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1] mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1] add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1] power(x, Nil) -> Cons(Nil, Nil) [1] mult(x, Nil) -> Nil [1] add0(x, Nil) -> x [1] goal(x, y) -> power(x, y) [1] The TRS has the following type information: power :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil mult :: Cons:Nil -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(x', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' goal(z, z') -{ 1 }-> power(x, y) :|: x >= 0, y >= 0, z = x, z' = y mult(z, z') -{ 1 }-> add0(x', mult(x', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 power(z, z') -{ 1 }-> mult(x', power(x', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x' power(z, z') -{ 1 }-> 1 + 0 + 0 :|: x >= 0, z = x, z' = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 goal(z0, z1) -> power(z0, z1) Tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) POWER(z0, Nil) -> c1 MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) MULT(z0, Nil) -> c3 ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) ADD0(z0, Nil) -> c5 GOAL(z0, z1) -> c6(POWER(z0, z1)) S tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) POWER(z0, Nil) -> c1 MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) MULT(z0, Nil) -> c3 ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) ADD0(z0, Nil) -> c5 GOAL(z0, z1) -> c6(POWER(z0, z1)) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, goal_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2, GOAL_2 Compound Symbols: c_2, c1, c2_2, c3, c4_1, c5, c6_1 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0, z1) -> c6(POWER(z0, z1)) Removed 3 trailing nodes: POWER(z0, Nil) -> c1 MULT(z0, Nil) -> c3 ADD0(z0, Nil) -> c5 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 goal(z0, z1) -> power(z0, z1) Tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) S tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, goal_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c4_1 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: goal(z0, z1) -> power(z0, z1) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) S tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c4_1 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) We considered the (Usable) Rules:none And the Tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) The order we found is given by the following interpretation: Polynomial interpretation : POL(ADD0(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [2] + x_1 + x_2 POL(MULT(x_1, x_2)) = [2] POL(Nil) = [2] POL(POWER(x_1, x_2)) = [3]x_2 POL(add0(x_1, x_2)) = [3] + [3]x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(mult(x_1, x_2)) = [2] + x_2 POL(power(x_1, x_2)) = [2] + [3]x_1 + [3]x_2 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) S tuples: MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c4_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) by POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)), POWER(z0, Nil)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)), POWER(z0, Nil)) S tuples: MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: MULT_2, ADD0_2, POWER_2 Compound Symbols: c2_2, c4_1, c_2 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) S tuples: MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: MULT_2, ADD0_2, POWER_2 Compound Symbols: c2_2, c4_1, c_2, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) by MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2(ADD0(z0, Nil), MULT(z0, Nil)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2(ADD0(z0, Nil), MULT(z0, Nil)) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2(ADD0(z0, Nil), MULT(z0, Nil)) K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c_1, c2_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2 S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2 K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c_1, c2_2, c2 ---------------------------------------- (51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MULT(z0, Cons(x1, Nil)) -> c2 We considered the (Usable) Rules:none And the Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2 The order we found is given by the following interpretation: Polynomial interpretation : POL(ADD0(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(MULT(x_1, x_2)) = [1] POL(Nil) = [1] POL(POWER(x_1, x_2)) = [1] + x_2 POL(add0(x_1, x_2)) = [1] + x_1 + x_2 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c2) = 0 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(mult(x_1, x_2)) = [1] + x_2 POL(power(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2 S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(x1, Nil)) -> c2 Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c_1, c2_2, c2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) by POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))), POWER(z0, Cons(x2, Nil))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) MULT(z0, Cons(x1, Nil)) -> c2 POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))), POWER(z0, Cons(x2, Nil))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) K tuples: POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) MULT(z0, Cons(x1, Nil)) -> c2 Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_1, c2_2, c2, c_2 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MULT(z0, Cons(x1, Nil)) -> c2 POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))), POWER(z0, Cons(x2, Nil))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, MULT_2, POWER_2 Compound Symbols: c4_1, c2_2, c_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, MULT_2, POWER_2 Compound Symbols: c4_1, c2_2, c_2, c_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) by MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)), MULT(z0, Cons(x2, Nil))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)), MULT(z0, Cons(x2, Nil))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)), MULT(z0, Cons(x2, Nil))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c_1, c2_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c_1, c2_2, c2_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) We considered the (Usable) Rules:none And the Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(ADD0(x_1, x_2)) = 0 POL(Cons(x_1, x_2)) = [1] + x_2 POL(MULT(x_1, x_2)) = [1] POL(Nil) = [1] POL(POWER(x_1, x_2)) = [1] + x_2 POL(add0(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(mult(x_1, x_2)) = [1] + x_2 POL(power(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c_1, c2_2, c2_1 ---------------------------------------- (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) by POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil)))) S tuples: ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c4_1, c_2, c2_2, c2_1, c_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) by ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil)))) ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) S tuples: MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c2_1, c_1, c4_1 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) by MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil)))) ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0)) S tuples: MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c_1, c4_1, c2_1 ---------------------------------------- (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil)))) by POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0)) POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil))) S tuples: MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c4_1, c2_1, c_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) by ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0)) POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil))) ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3)))) S tuples: MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3)))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c2_1, c_1, c4_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0)) by MULT(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Cons(z2, Nil))) -> c2(ADD0(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(y1, Cons(y2, Cons(y3, y4))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2)) power(z0, Nil) -> Cons(Nil, Nil) mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2)) mult(z0, Nil) -> Nil add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2)) add0(z0, Nil) -> z0 Tuples: POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2)))) MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil))) ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3)))) MULT(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Cons(z2, Nil))) -> c2(ADD0(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(y1, Cons(y2, Cons(y3, y4))))) S tuples: MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2)))) ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3)))) K tuples: MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) Defined Rule Symbols: power_2, mult_2, add0_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c_2, c2_2, c_1, c4_1, c2_1